Drell–Yan production at small q _T , transverse parton distributions and the collinear anomaly

Abstract

Using methods from effective field theory, an exact all-order expression for the Drell–Yan cross section at small transverse momentum is derived directly in q _T space, in which all large logarithms are resummed. The anomalous dimensions and matching coefficients necessary for resummation at NNLL order are given explicitly. The precise relation between our result and the Collins–Soper–Sterman formula is discussed, and as a by-product the previously unknown three-loop coefficient A ⁽³⁾ is obtained. The naive factorization of the cross section at small transverse momentum is broken by a collinear anomaly, which prevents a process-independent definition of x _T -dependent parton distribution functions. A factorization theorem is derived for the product of two such functions, in which the dependence on the hard momentum transfer is separated out. The remainder factors into a product of two functions of longitudinal momentum variables and $x_{T}^{2}$, whose renormalization-group evolution is derived and solved in closed form. The matching of these functions at small x _T onto standard parton distributions is calculated at $\mathcal{O}(\alpha_{s})$, while their anomalous dimensions are known to three loops.